Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 31J / b / Solution

A function $f:C\to \mathbb{R}$ is strictly convex when, for distinct $x,y\in C$ and $0<t<1$,

Suppose distinct points $x$ and $y$ were both minimizers, with minimum value $m$. Since $C$ is a convex set, their midpoint belongs to $C$, while strict convexity givesa contradiction. Thus any minimizer is unique, which is the uniqueness of a minimizer of a strictly convex function.